"The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun." http://en.wikipedia.org/wiki/Amenable_group

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I will add a quote from Runde, V. (2002), Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer, ISBN 9783540428527, p.34. (Notice that the Wikipedia article says in a footnote: "Day's first published use of the word is in his abstract for an AMS summer meeting in 1949, Means on semigroups and groups, Bull. A.M.S. 55 (1949) 1054–1055. Many text books on amenabilty, such as Volker Runde's, suggest that Day chose the word as a pun.")

Amenable (discrete) groups were first considered by J. von Neumann, albeit under a different name ([Neu]). The first to use the adjective "amenable" was M. M. Day in [Day], apparently with a pun in mind: These groups $G$ are called amenable because they have an invariant mean on $L^\infty(G)$, but also since they are particularly pleasant to deal with and thus are truly amenable - just in the sense of that adjective in colloquial English.

So the above explanations confirms what André Nicolas wrote in his comment